论尼罗河问题罗纳德·费希尔爵士
On the Nile Problem by Sir Ronald Fisher
摘要：
Ronald Fisher的Nile问题可以解释为当最小充分统计量不完全时对一个特殊的曲指数族进行统计推断的问题。问题本身和它的版本一般弯曲指数家庭提出了一个数学统计的挑战：研究从属统计量的子代数内的$σ$ -代数的（不完全）最小充分统计量和密切相关的问题的结构UMVUE。本文提出了一种新的方法，特别是，证明了在经典的尼罗河问题没有统计受到温和的自然条件是UMVUE。该方法几乎解决了UMVUE存在的一个老问题。该方法是纯统计的（与分析），适用于任何拥有从属统计量的家族。它补充了一个分析方法，只使用一阶辅助（因此，工作时存在的辅助子代数是一个开放的问题），并适用于弯曲的指数族与多项式的规范参数的限制，其中尼罗河问题是一个特例。
英文摘要：
The Nile problem by Ronald Fisher may be interpreted as the problem of making statistical inference for a special curved exponential family when the minimal sufficient statistic is incomplete. The problem itself and its versions for general curved exponential families pose a mathematical-statistical challenge: studying the subalgebras of ancillary statistics within the $σ$-algebra of the (incomplete) minimal sufficient statistics and closely related questions of the structure of UMVUEs. In this paper a new method is developed that, in particular, proves that in the classical Nile problem no statistic subject to mild natural conditions is a UMVUE. The method almost solves an old problem of the existence of UMVUEs. The method is purely statistical (vs. analytical) and works for any family possessing an ancillary statistic. It complements an analytical method that uses only the first order ancillarity (and thus works when the existence of ancillary subalgebras is an open problem) and works for curved exponential families with polynomial constraints on the canonical parameters of which the Nile problem is a special case.
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